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A219844 T(n,k)=Number of -k..k arrays of length n whose end-around centered second difference is a constant times that array. 1

%I

%S 3,5,5,7,9,9,9,13,23,13,11,17,43,33,3,13,21,69,61,5,17,15,25,101,97,7,

%T 45,3,17,29,139,141,9,85,5,13,19,33,183,193,11,137,7,33,9,21,37,233,

%U 253,13,201,9,61,23,5,23,41,289,321,15,277,11,97,43,9,3,25,45,351,397,17,365,13

%N T(n,k)=Number of -k..k arrays of length n whose end-around centered second difference is a constant times that array.

%C The empirical formula below depends on eigenvectors of the system existing only with the discovered periods of 1, 2, 3, 4 and 6

%C The eigenvectors are sampled sines and cosines, and so the formula depends on no sampling x(j) of sin(j*2Pi/n+offset) existing with exclusively rational ratios of the x(j) except for those already found for n=1,2,3,4 or 6

%C The empirical formula dependency on k then describes how those existing cycles can be populated with new values

%H R. H. Hardin, <a href="/A219844/b219844.txt">Table of n, a(n) for n = 1..476</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Crystallographic_restriction_theorem">Crystallographic restriction theorem</a>.

%F .Empirical: t(n,k) { #pseudocode

%F . if (n modulo 12 = 0) return 10*k^2+14*k+1;

%F . if (n modulo 12 = 1 || n modulo 12 = 5 || n modulo 12 = 7 || n modulo 12 = 11) return 2*k+1;

%F . if (n modulo 12 = 2 || n modulo 12 = 10) return 4*k+1;

%F . if (n modulo 12 = 3 || n modulo 12 = 9) return 3*k^2+5*k+1;

%F . if (n modulo 12 = 4 || n modulo 12 = 8) return 4*k^2+8*k+1;

%F . if (n modulo 12 = 6) return 6*k^2+10*k+1;

%F .}

%e Table starts

%e ..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37

%e ..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73

%e ..9.23..43..69.101.139.183.233.289..351..419..493..573..659..751..849..953.1063

%e .13.33..61..97.141.193.253.321.397..481..573..673..781..897.1021.1153.1293.1441

%e ..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37

%e .17.45..85.137.201.277.365.465.577..701..837..985.1145.1317.1501.1697.1905.2125

%e ..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37

%e .13.33..61..97.141.193.253.321.397..481..573..673..781..897.1021.1153.1293.1441

%e ..9.23..43..69.101.139.183.233.289..351..419..493..573..659..751..849..953.1063

%e ..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73

%e ..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37

%e .25.69.133.217.321.445.589.753.937.1141.1365.1609.1873.2157.2461.2785.3129.3493

%e ..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37

%e Some solutions for n=6 k=5, alternating array and its second difference

%e ..2...-2....3...-3....1...-1....0....0....3...-9....3..-12....1...-3....2...-6

%e ..2...-2...-1....1....0....0....4...-4....2...-6...-3...12...-3....9...-4...12

%e ..0....0...-4....4...-1....1....4...-4...-5...15....3..-12....2...-6....2...-6

%e .-2....2...-3....3...-1....1....0....0....3...-9...-3...12....1...-3....2...-6

%e .-2....2....1...-1....0....0...-4....4....2...-6....3..-12...-3....9...-4...12

%e ..0....0....4...-4....1...-1...-4....4...-5...15...-3...12....2...-6....2...-6

%e Some solutions for n=12 k=5, alternating array and its second difference

%e .-3....3...-3....9...-5...10...-4...12....4...-4...-3....3....2...-8....1...-3

%e ..0....0....0....0...-2....4....2...-6...-1....1...-2....2...-2....8....2...-6

%e ..3...-3....3...-9....5..-10....2...-6...-5....5....1...-1....2...-8...-3....9

%e ..3...-3...-3....9....2...-4...-4...12...-4....4....3...-3...-2....8....1...-3

%e ..0....0....0....0...-5...10....2...-6....1...-1....2...-2....2...-8....2...-6

%e .-3....3....3...-9...-2....4....2...-6....5...-5...-1....1...-2....8...-3....9

%e .-3....3...-3....9....5..-10...-4...12....4...-4...-3....3....2...-8....1...-3

%e ..0....0....0....0....2...-4....2...-6...-1....1...-2....2...-2....8....2...-6

%e ..3...-3....3...-9...-5...10....2...-6...-5....5....1...-1....2...-8...-3....9

%e ..3...-3...-3....9...-2....4...-4...12...-4....4....3...-3...-2....8....1...-3

%e ..0....0....0....0....5..-10....2...-6....1...-1....2...-2....2...-8....2...-6

%e .-3....3....3...-9....2...-4....2...-6....5...-5...-1....1...-2....8...-3....9

%o (PARI) t(n,k)={1+[[10,14],[0,2],[0,4],[3,5],[4,8],[0,2],[6,10]][7-abs(n%12-6)]*[k^2,k]~} \\ Using the pseudocode given as formula. - _M. F. Hasler_, Feb 20 2013

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_ Feb 19 2013

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Last modified October 1 08:39 EDT 2020. Contains 337442 sequences. (Running on oeis4.)